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NCERT Exemplar Class 11 Maths solutions chapter 5 covers Complex numbers and its applications. Complex numbers are something which we have been studying over the years with any ‘real number’ being said to be a complex number, which can be denoted as a variable or alphabet in simpler terms. The other number, along with the real number, is a part of the complex number, is called the ‘imaginary number.’ Both of them appear together in a problem. Terms such as integer, conjugate, square root, polar, etc. will be studied in NCERT Exemplar Class 11 Maths solutions chapter 5, that will be utilised to find results for these complex number problems.
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Question:1
For a positive integer n, find the value of
Answer:
Question:8
If, then show that where b?R, representing z in the complex plane is a circle.
Answer:
This equation is a equation of circle
Question:9
Answer:
The equation obtained is that of a circle. Hence, locus of z is a circle.
Question:10
Show that the complex number z, satisfying the condition lies on a circle.
Answer:
Let z=x+iy
The equation obtained represents the equation of a circle
Question:11
Answer:
Comparing real and imaginary parts
And
Substituting the value of y in
Hence,
Hence,
Question:12
Answer:
We have
Substituting we get
Comparing real and imaginary parts,
Substituting the value of y in
Hence,
Thus,
Question:14
Show that represents a circle. Find its centre and radius.
Answer:
Substituting , we get
Thus, the centre of circle is and radius is
Question:15
If is a purely imaginary number, then find the value of .
Answer:
Let
is purely imaginary
Hence,
Question:16
z1 and z2 are two complex numbers such that and , then show that .
Answer:
Let and
Given that
And
,
Question:17
If |z1| = 1 (z1 ≠ –1) and then show that the real part of z2 is zero
Answer:
Let
Since,
Therefore, the real part of is zero
Question:18
If z1, z2 and z3, z4 are two pairs of conjugate complex numbers, then find
Answer:
and are conjugate complex numbers.
The negative side of the real axis
Similarly,
Question:22
Find the complex number satisfying the equation .
Answer:
Substituting we get
1+y = 0
y = -1
Hence,
Question:25
i) For any two complex numbers z1, z2 and any real numbers a, b,
ii) The value of is
iii) The number is equal to .......
iv) The sum of the series upto 1000 terms is ...
v) Multiplicative inverse of 1 + i is ................
vi) If are complex numbers such that is a real number, then =
vii) is ....
viii) If then the greatest and least values of |z+1| are ..... and .....
ix) if then the locus of z is .......
x) If and then z=
Answer:
i)
ii)
iii)
iv)
v)
vi) Let
If is real then
So,
vii)
viii)
The gratest value is 6 and the least value of is 0
ix)
Let
which represents an equation of a circle
x) and
Let
Question:26
Answer:
(i) Comparison of two purely imaginary complex numbers is not possible. However, the two purely real complex number can be compared. So, it is false.
(ii) Let
which rotates at angle of 180. So, it is ‘false’.
(iii) Let
The value of is minimum when
Hence, it is true.
iv) Let
Given that
which is a straight line slope=1
Now, equation of line through the point 1,0and 0,1
whose slope=-1
Multiplication of the slopes of two lines =-1*1=-1
So, they are perpendicular. Hence, it is true.
v)Let
Since, real part is 0
which is real Hence, it is False.
vi)
Let
Hence, it is true.
vii) Let and
Squaring both sides, we get )
Again squaring on both sides we get
Hence, it is false.
(viii) Since, every real number is a complex number. So, 2 is a complex number. Hence, it is false.
Question:27
a) The polar form of is | i) Perpendicular bisector of segment joining (-2,0) and (2,0) |
b)The amplitude of is | ii) On or outside the circle having centre at (0,-4) and radius 3. |
c) If then locus of z is | iii) |
d)If then locus of z is | iv)Perpendicular bisector of segment joining (0,-2) and (0,2) |
e) Region represented by is | v) |
f) Region represented by is | vi) On or outside the circle having centre at (-4,0) and radius 3units. |
g)Conjugate of lies in | vii) First Quadrant |
h) Reciprocal of 1-i lies in | viii) Third Quadrant |
Answer:
a)Given
Polar form of z .
And
Since
Polar form of z
b) Given that
Here argument z
So,
Since, x<0 and y>0
c)
which represents equation of-y axis and is perpendicular to the line joining the points (-2,0) and (2,0)
d)
which is the equation of x-axis and is perpendicular to the line segment joining (0,-2)and (0,2)
e)
which represents a circle on or outside having centre (0,-4)
f)
Let Then,
which is a circle having centre (-4,0) and and is on the circle.
g) Let
which lies in third quadrant..
h) Given that z=1-i
Reciprocal of
which lies in first quadrant.
Question:35
and are conjugate to each other for:
A.
B.
C.
D. No value of x
Answer:
The answer is the option (c).
Comparing the real and imaginary part, we get
The above value is not satisfied by tan x = 1. Hence no value of x is possible.
Question:36
The real value of α for which the expression is purely real is:
A.
B.
C.
D. None of these
Answer:
The answer is the option (c).
Let
Question:37
If z = x + iy lies in the third quadrant, the also lies in the third quadrant if
A. x > y > 0
B. x < y < 0
C. y < x < 0
D. y > x > 0
Answer:
The answer is the option (b).
If z lies in the third quadrant,So,
When z lies in third quadrant then will also be in third quadrant.
Hence, b is correct..
Question:38
The value of is equivalent to
A.
B. |z – 3|
C.
D. None of these
Answer:
The answer is the option (a).
A real value of x satisfies the equation () , if
A. 1
B. –1
C. 2
D. –2
Answer:
The answer is the option (a).
Multiplying eqn (i )and (ii )we get
Question:41
Which of the following is correct for any two complex numbers ?
A.
B.
C.
D.
Answer:
The answer is the option (a).
Hence, option a is correct.
Question:42
Answer:
The answer is the option (b).
If z rotated through an angle of about the origin in clockwise direction .
Then the new position
Hence, the correct option is (b)
Question:43
Let , then is a non-real complex number if:
A.
B.
C.
D.
Answer:
The answer is the option (d).
is a non real complex number if ,
Hence, d is correct
Question:44
If , then
A.
B.
C.
D.
Answer:
The answer is the option (d).
Squaring both sides,
Hence, d is correct
Question:45
The complex number z which satisfies the condition lies on
A. circle
B. the x-axis
C. the y-axis
D. the line
Answer:
The answer is the option (b).
Hence, b is correct
Question:46
If z is a complex number, then
A.
B.
C.
D.
Answer:
. The answer is the option (b).
Hence, b is correct
Question:47
B. z2=z1
C.
D.
Answer:
The answer is the option (c).
Since,
So,
Squaring both sides,
Question:48
The real value of θ for which the expression is a real number is:
A.
B.
C.
D. none of these
Answer:
On solving we get
If z is real number then
Question:49
The value of when is:
A. 0
B.
C. π
D. none of these
Answer:
The answer is the option (c).
Since, the point (-x,0)lies on the negative side of the real axis , principal argument (z)=π
Hence, option c is correct
Question:50
If where , then is
A.
B.
C.
D. none of these
Answer:
The answer is the option (a).
Given that
Now
Hence, option a is correct
Various laws need to be followed and derived by a mathematician when it comes to multiplying the two complex numbers. All of them are named over the features of the specific formula derivation. Ahead, you will also learn about the Modulus and Conjugate of complex numbers and how to solve sums with their help. There is an Argand and Polar representation in the chapter, relating to graphical presentations. A point needs to be derived at the ‘x’ and ‘y’ axis to form an argand or complex plane.
Students, with the help of NCERT Exemplar Class 11 Maths solutions chapter 5, will not face any issues when trying to solve such problems. It will also help them score better in exams.
All the concepts have been covered in NCERT Exemplar solutions for Class 11 Maths chapter 5. By using NCERT Exemplar Class 11 Maths chapter 5 solutions PDF Download function, students can access quality study material that is effectively constructed by experts for the best learning experience.
Belonging to the algebraic part of the subject, it is always fun for the students to solve these problems, once they are clear with the concept. NCERT Exemplar Class 11 Maths chapter 5 solutions will help in understanding the said concept and give examples in how to apply them to questions of any kind.
Other than that, in geometric applications, complex numbers are also very useful, for students in their near future. Students, through NCERT Exemplar Class 11 Maths solutions chapter 5, will also learn graphs in an advanced manner with newer placements and some difficult problems, which would make them sharper and clever.
· Class 11 Maths NCERT Exemplar solutions chapter 5 has detailed that complex numbers, their operations, square root, power, identities, representation of complex numbers and quadratic equations are important topics which students should pay extra attention to.
· In NCERT Exemplar Class 11 Maths chapter 5 solutions, students will also learn about quadratic equations. A quadratic equation is an equation where rearrangement of a particular equation is done via a standard form, where x is the unknown number, whereas, a, b and c, are known numbers. An equation with four numbers is said to be quadratic equations.
Check Chapter-Wise NCERT Solutions of Book
Chapter-1 | |
Chapter-2 | |
Chapter-3 | |
Chapter-4 | |
Chapter-5 | Complex Numbers and Quadratic equations |
Chapter-6 | |
Chapter-7 | |
Chapter-8 | |
Chapter-9 | |
Chapter-10 | |
Chapter-11 | |
Chapter-12 | |
Chapter-13 | |
Chapter-14 | |
Chapter-15 | |
Chapter-16 |
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Yes, for those who want to score high in their school exams and also want to do better in entrance exams, practising NCERT questions is a must.
Yes, our experts have solved every question in the most detail-oriented way in NCERT Exemplar Class 11 Maths solutions Chapter 5.
In this chapter, the complex numbers are explained, their properties, operations on complex numbers along with quadratic equations.
Yes, the solutions can be downloaded using the NCERT Exemplar Class 11 Maths Chapter 5 solutions PDF Download function, as one can get the offline PDF after clicking on the download link.
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An AWS Solution Architect is someone who specializes in developing and implementing cloud computing systems. He or she has a good understanding of the various aspects of cloud computing and can confidently deploy and manage their systems. He or she troubleshoots the issues and evaluates the risk from the third party.
An Azure Administrator is a professional responsible for implementing, monitoring, and maintaining Azure Solutions. He or she manages cloud infrastructure service instances and various cloud servers as well as sets up public and private cloud systems.
Careers in computer programming primarily refer to the systematic act of writing code and moreover include wider computer science areas. The word 'programmer' or 'coder' has entered into practice with the growing number of newly self-taught tech enthusiasts. Computer programming careers involve the use of designs created by software developers and engineers and transforming them into commands that can be implemented by computers. These commands result in regular usage of social media sites, word-processing applications and browsers.
A Product Manager is a professional responsible for product planning and marketing. He or she manages the product throughout the Product Life Cycle, gathering and prioritising the product. A product manager job description includes defining the product vision and working closely with team members of other departments to deliver winning products.
Individuals in the information security manager career path involves in overseeing and controlling all aspects of computer security. The IT security manager job description includes planning and carrying out security measures to protect the business data and information from corruption, theft, unauthorised access, and deliberate attack
An Automation Test Engineer job involves executing automated test scripts. He or she identifies the project’s problems and troubleshoots them. The role involves documenting the defect using management tools. He or she works with the application team in order to resolve any issues arising during the testing process.
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